What's the best way to start off Skyrim if you want to be a warrior, a thief, or a mage?

Do what you want. Choose the perks you believe will be best suited and have fun. Don't think you're playing the game wrong, either. Here's a paraphrase from the manual for the GOTY edition of Morrowind.

"No matter what your preference, there's no right or wrong way to play."

Personally i play as a mage but the possibilities are endless, not really a fan of the thief its a pretty boring class in my opinion, if i were to choose a warrior it would be a badass heavy armour wearing orc , if you go for a mage choose a high elf or bosmer, but its skyrim you evolve your character as you see fit.